On nonexistence of splash singularities for the $\alpha$-SQG patches

Abstract

In this paper, we consider patch solutions to the $\alpha$-SQG equation and derive new criteria for the absence of splash singularity where different patches or parts of the same patch collide in finite time. Our criterion refines a result due to Gancedo and Strain \cite{GS}, providing a condition on the growth of curvature of the patch necessary for the splash and an exponential in time lower bound on the distance between patches with bounded curvature.

arXiv comment12 pages; corrected some exponents

References in corpus (3)

• Global regularity for a modified critical dissipative quasi-geostrophic equation
• Absence of splash singularities for SQG sharp fronts and the Muskat problem
• A self-similar cascade of instabilities in the surface quasigeostrophic system